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ON $\mu _{n}$-ACTIONS ON K3 SURFACES IN POSITIVE CHARACTERISTIC

Published online by Cambridge University Press:  05 September 2022

YUYA MATSUMOTO*
Affiliation:
Department of Mathematics Faculty of Science and Technology Tokyo University of Science 2641 Yamazaki, Noda, Chiba 278-8510 Japan matsumoto.yuya.m@gmail.com matsumoto_yuya@ma.noda.tus.ac.jp
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Abstract

In characteristic $0$ , symplectic automorphisms of K3 surfaces (i.e., automorphisms preserving the global $2$ -form) and non-symplectic ones behave differently. In this paper, we consider the actions of the group schemes $\mu _{n}$ on K3 surfaces (possibly with rational double point [RDP] singularities) in characteristic p, where n may be divisible by p. We introduce the notion of symplecticness of such actions, and we show that symplectic $\mu _{n}$ -actions have similar properties, such as possible orders, fixed loci, and quotients, to symplectic automorphisms of order n in characteristic $0$ . We also study local $\mu _n$ -actions on RDPs.

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© The Author(s), 2022. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

1 Introduction

K3 surfaces are proper smooth surfaces X with $\Omega ^2_X \cong \mathcal {O}_X$ and $H^1(X, \mathcal {O}_X) = 0$ . The first condition implies that X admits an everywhere nonvanishing $2$ -form, and such a $2$ -form is unique up to scalar. An automorphism of a K3 surface is called symplectic if it preserves the global $2$ -form. It is known that symplectic and non-symplectic automorphisms behave very differently.

For example, Nikulin [Reference MukaiNi, §§4 and 5] proved that quotients of K3 surfaces in characteristic $0$ by a symplectic action of a finite group G has only rational double points (RDPs) as singularities and that the minimal resolutions of the quotients are again K3 surfaces. Moreover, he determined the number of fixed points (which are always isolated) if G is cyclic. To the contrary, the quotients by non-symplectic actions of finite groups are never birational to K3 surfaces; instead, they are birational to either Enriques surfaces or rational surfaces.

These results hold in characteristic $p> 0$ provided p does not divide the order of G (see Theorem 5.1), but are no longer true for order p automorphisms in characteristic p. In this case, the notion of symplecticness is useless, since any such automorphism is automatically symplectic (since there are no nontrivial pth root of unity in characteristic p) and, for small p, there exist examples of automorphisms with non-K3 quotients (see [Reference Dolgachev and KeumDK1], [Reference Dolgachev and KeumDK2]).

In this paper, we consider actions of the finite group schemes $\mu _{n}$ (n may be divisible by p) on RDP K3 surfaces, by which we mean surfaces with RDP singularities whose minimal resolutions are K3 surfaces. It is essential to allow RDPs since smooth K3 surfaces never admit actions of $\mu _p$ (see Remark 2.2). We introduce the notion of symplecticness and fixed points of such actions (Definitions 2.5 and 2.7). Then we prove the following properties, which are parallel to the properties of automorphisms of order not divisible by the characteristic.

Theorem 1.1 (Theorems 6.1 and 6.2)

Let X be an RDP K3 surface in characteristic p, equipped with a $\mu _{n}$ -action. If the action is symplectic, then the quotient $X/\mu _{n}$ is an RDP K3 surface. If $n = p$ and the action is non-symplectic, then the quotient $X/\mu _{p}$ is an RDP Enriques surface if the action is fixed-point-free (which is possible only if $p = 2$ ), and otherwise it is a rational surface.

Theorem 1.2 (Theorems 7.1 and 8.2)

  • There exists an RDP K3 surface X in characteristic p equipped with a $\mu _{p}$ -action if and only if $p \leq 19$ .

  • If X is an RDP K3 surface X in characteristic p equipped with a $\mu _{n}$ -action, then $\phi (n) \leq 20$ , in particular $n \leq 66$ . Moreover, for each p, we determine the set of n for which such an action exists.

  • For each p, there exists an RDP K3 surface X in characteristic p equipped with a symplectic $\mu _{n}$ -action if and only if $n \leq 8$ , and we determine the number of fixed points.

To prove the main results, we first study (in §§3 and 4) $\mu _n$ -actions on local rings of surfaces at smooth points and RDPs. We define the notion of symplecticness of such actions (Definitions 3.1 and 4.1) and prove the following result.

Theorem 1.3 (Theorem 4.6 and Propositions 4.7 and 4.9)

Let X be the localization at a closed point w of an RDP surface in characteristic p equipped with a $\mu _{p}$ -action. Let ${\pi } : {X}\to {X/\mu _p}$ be the quotient morphism.

  • If w is not fixed by the action, then $\pi (w)$ is either a smooth point or an RDP.

  • If w is fixed and the action is symplectic at w, then w is an isolated fixed point and $\pi (w)$ is an RDP.

  • If w is an isolated fixed point and the action is non-symplectic at w, then $\pi (w)$ is a non-RDP singularity.

We classify the possible actions in the non-fixed case (Table 3) and the symplectic case (Table 4).

Moreover, we also give a partial classification of local $\mu _{p^e}$ - and $\mu _n$ -actions (Propositions 4.12 and 4.13) and a complete classification of local symplectic $\mu _n$ -actions (Proposition 4.14). We hope that these local results would have applications other than K3 surfaces.

The results on $\mu _n$ -quotients, orders of symplectic $\mu _n$ -actions, and orders of $\mu _n$ -actions on K3 surfaces are discussed in §§68, respectively.

In §9, we give several examples of $\mu _{n}$ -actions on K3 surfaces.

Throughout the paper, we work over an algebraically closed field k of $\operatorname {\mathrm {char}} k = p \geq 0$ . Varieties are separated integral k-schemes of finite type (not necessarily proper or smooth), and surfaces are two-dimensional varieties. We denote the smooth locus of a variety X by $X^{\mathrm {sm}}$ .

2 Preliminaries

2.1 K3 surfaces and rational double points

RDP singularities of surfaces are precisely the canonical surface singularities that are not smooth. They are classified into types $A_n$ ( $n \geq 1$ ), $D_n$ ( $n \geq 4$ ), and $E_n$ ( $n = 6,7,8$ ) by their dual graph of the exceptional curves of the minimal resolution, which are Dynkin diagrams of type $A_n$ , $D_n$ , or $E_n$ . The number n is equal to the number of the exceptional curves, and we say that the RDP is of index n. The dual graph determines the formal isomorphism class of an RDP except in certain cases in characteristics $2,3,5$ . For the exceptional cases, we use Artin’s notation $D_n^r$ and $E_n^r$ (see [Reference ArtinA2]).

We recall the classification, given by Bombieri and Mumford [Reference Bombieri and MumfordBM2], of proper smooth surfaces X with numerically trivial canonical divisor $K_X$ : they consist of four classes, with the characterizing properties as reported in Table 1. Here, $b_i = \dim H_{\mathrm {\acute et}}^i(X, \mathbb Q_l)$ is the ith l-adic Betti number for an auxiliary prime $l \neq \operatorname {\mathrm {char}} k$ . Enriques and (quasi-)hyperelliptic surfaces in characteristics $2$ and $3$ may have unusual values of $\dim H^1(\mathcal {O}_X)$ and $\operatorname {\mathrm {ord}}(K_X)$ .

Table 1 Surfaces with numerically trivial canonical divisors.

The distinction between hyperelliptic and quasi-hyperelliptic surfaces is not important in this paper. Furthermore, the choice of the origin of an abelian surface is not important.

Definition 2.1. RDP surfaces are surfaces that have only RDPs as singularities (if any). Hence, any smooth surface is an RDP surface by definition.

RDP K3 surfaces are proper RDP surfaces whose minimal resolutions are (smooth) K3 surfaces. We similarly define RDP abelian, RDP Enriques, and RDP (quasi-)hyperelliptic surfaces.

However, since abelian surfaces and (quasi-)hyperelliptic surfaces do not admit smooth rational curves, any RDP abelian or RDP (quasi-)hyperelliptic surface is smooth.

Remark 2.2. Smooth K3 surfaces in characteristic $p> 0$ admit no nontrivial global vector fields ([Reference OgusRS, Th. 7], [Reference NikulinNy1, Cor. 3.5]), and hence admit no nontrivial actions of $\mu _p$ (or $\alpha _p$ ). However, RDP K3 surfaces may admit such actions.

Proposition 2.3. For any RDP surface X, the pullback by the morphism $X^{\mathrm {sm}} \cong \tilde X \setminus E \hookrightarrow \tilde X$ to the minimal resolution $\tilde X$ of X induces an isomorphism $H^0(X^{\mathrm {sm}}, (\Omega ^2_X)^{\otimes n}) \cong H^0(\tilde X, (\Omega ^2_{\tilde X})^{\otimes n})$ , where E is the exceptional divisor. Nonvanishing forms on one side correspond to nonvanishing ones on the other side.

Proof. This follows from the following local version applied repeatedly.

Proposition 2.4. Let $(A,\mathfrak {m})$ be the localization at a closed point of an RDP surface. Then $H^0(\operatorname {\mathrm {Spec}} A \setminus \{\mathfrak {m}\}, \Omega ^2_{A/k})$ is a free A-module of rank 1. If A is smooth, then this space is isomorphic to $H^0(\operatorname {\mathrm {Spec}} A, \Omega ^2_{A/k})$ . If A is an RDP and $(A',\mathfrak {m}')$ is the localization at a closed point of $\operatorname {\mathrm {Bl}}_{\mathfrak {m}} A$ , then any generator of the above space extends to a generator of $H^0(\operatorname {\mathrm {Spec}} A' \setminus \{\mathfrak {m}'\}, \Omega ^2_{A'/k})$ .

Proof. If A is smooth, then $\Omega ^2_{A/k}$ is free of rank $1$ and the assertion is clear. Suppose A is an RDP. Then it is a hypersurface isolated singularity, and it is well known that for such singularities, the canonical divisor is trivial, and then the former assertion follows. Since an RDP is a canonical singularity, the pullback of the canonical divisor to $\operatorname {\mathrm {Bl}}_{\mathfrak {m}} A$ is also trivial, and hence the latter assertion follows.

2.2 Group schemes of multiplicative type

Recall that we are working over an algebraically closed field k.

We consider finite commutative group schemes G of multiplicative type over k. This means that G is of the form $\prod _j \mu _{n_j}$ for some positive integers $n_j$ . The Cartier dual $G^{\vee } = \operatorname {\mathrm {\mathcal {H}\mathit {om}}}(G, \mathbb G_m)$ of G is a finite étale group scheme and can be identified with the finite group $G^{\vee }(k)$ of k-valued points. Using this finite commutative group $G^{\vee }$ , we have the following explicit description: $G = \operatorname {\mathrm {Spec}} k[t_{i}]_{i \in G^{\vee }} / (t_{i} t_{j} - t_{i+j}, t_0 - 1)$ , with the group operations $m \colon G \times G \to G$ , $e \colon \operatorname {\mathrm {Spec}} k \to G$ , $i \colon G \to G$ given by $m^*(t_{i}) = t_{i} \otimes t_{i}$ , $e^*(t_{i}) = 1$ , $i^*(t_{i}) = t_{-i}$ .

An action $\alpha \colon G \times \operatorname {\mathrm {Spec}} B \to \operatorname {\mathrm {Spec}} B$ corresponds, via $\alpha ^*(b) = \sum _{i \in G^{\vee }} t_{i} \otimes \operatorname {\mathrm {pr}}_{i}(b)$ , to decompositions $B = \bigoplus _{i \in G^{\vee }} B_i$ to k-vector subspaces satisfying $B_i B_j \subset B_{i+j}$ . We say an element b or a subset of $B_i$ to be homogeneous of weight i and we write $\operatorname {\mathrm {wt}}(b) = i$ .

Such a decomposition $B = \bigoplus _{i} B_i$ naturally extends to a decomposition $\Omega ^*_{B/k} = \bigoplus _i (\Omega ^*_{B/k})_i$ satisfying $d(B_i) \subset (\Omega ^1_{B/k})_i$ and $(\Omega ^*_{B/k})_i (\Omega ^*_{B/k})_j \subset (\Omega ^*_{B/k})_{i+j}$ .

If G acts on a scheme X that is not necessarily affine but admits a covering by G-stable affine open subschemes (which is the case if, e.g., X is quasi-projective or G is local), then the G-action admits a quotient ${\pi }:{X} \to {X/G}$ , and induces decompositions $\pi _* \mathcal {O}_X = \bigoplus _{i} (\pi _* \mathcal {O}_X)_i$ , $\pi _* \Omega ^*_{X/k} = \bigoplus _{i} (\pi _* \Omega ^*_{X/k})_i$ , and $H^0(X, (\Omega ^*_{X/k})^{\otimes n}) = \bigoplus _i (H^0(X, (\Omega ^*_{X/k})^{\otimes n}))_i$ , compatible with multiplications.

If $\operatorname {\mathrm {char}} k$ does not divide the order of $G^{\vee }$ , then $B_i$ are the eigenspaces for the action of $G(k)$ with eigenvalues $i \in G^{\vee }(k) = \operatorname {\mathrm {Hom}}(G(k),k^*)$ .

If $\operatorname {\mathrm {char}} k = p> 0$ and $G^{\vee }$ is cyclic of order p (hence $G \cong \mu _p = \operatorname {\mathrm {Spec}} k[t_1]/(t_1^p - 1)$ for a choice of a generator $1$ of $G^{\vee }$ ), then giving such a decomposition is also equivalent to giving a k-derivation D on B of multiplicative type (i.e., $D^p = D$ ) under the correspondence $B_i = B^{D = i} = \{b \in B \mid D(b) = ib \}$ (this correspondence depends on the choice of a generator $1$ of $G^{\vee }$ ). Moreover, D extends to a k-linear endomorphism on $\Omega ^*_{B/k}$ satisfying $D(df) = d(D(f))$ , $D^p = D$ , and the Leibniz rule $D(\omega \wedge \eta ) = \omega \wedge D(\eta ) + D(\omega ) \wedge \eta $ .

Now, we generalize the notion of symplecticness of automorphisms to actions of group schemes like $\mu _n$ .

Definition 2.5. Let G be a finite group scheme of multiplicative type. Let X be either an abelian surface or an RDP K3 surface, equipped with an action of G. We say that the action is symplectic if the one-dimensional space $H^0(X^{\mathrm {sm}}, \Omega ^2_{X/k})$ with respect to the action of G is of weight $0$ .

Remark 2.6. Under the assumptions of Definition 2.5, suppose G is reduced. Equivalently, this means that G is a constant group scheme corresponding to a finite abelian group of order prime to p. Then, by Proposition 2.3, our symplecticness is equivalent to the symplecticness of the induced G-action on the minimal resolution $\tilde {X}$ in the usual sense (i.e., preserving the global $2$ -form). This suggests that our definition of the symplecticness of $\mu _n$ -actions is a natural generalization of that of $\mathbb Z/m\mathbb Z$ -actions (order m automorphisms) for m not divisible by $\operatorname {\mathrm {char}} k$ .

On the other hand, if $G = \mathbb Z/p\mathbb Z$ (which does not belong to the class considered in Definition 2.5), then any action of G preserves the global $2$ -form, since there are no nontrivial pth roots of unity. Thus, the usual definition of symplecticness is useless in this case. We do not know whether there is a useful notion of symplecticness in a larger class of group schemes containing $\mathbb Z/p\mathbb Z$ or $\alpha _p$ .

2.3 Derivations of multiplicative type

In this section, assume that $\operatorname {\mathrm {char}} k = p> 0$ .

Recall that, given an action of a group scheme G on a scheme X, the fixed point scheme $X^G \subset X$ is characterized by the property $X^G(T) = \operatorname {\mathrm {Hom}}_G(T,X)$ for any k-scheme T equipped with the trivial G-action. If $G = \mu _p$ and D is the corresponding derivation, we write $\operatorname {\mathrm {Fix}}(D) = X^G$ and also call it the fixed locus of D.

Definition 2.7. We say that a closed point $w \in X$ is fixed by the $\mu _{n}$ -action, or by the corresponding derivation if $n = p$ , if $w \in X^{\mu _{n}}$ .

Proposition 2.8. Let k be an algebraically closed field. Let $X = \operatorname {\mathrm {Spec}} B$ be a Noetherian affine k-scheme equipped with a $\mu _{p^e}$ -action. For each closed point $w \in X$ , the assertions (1)–(4) are equivalent. If $e = 1$ and D is the corresponding derivation, then the assertions (1)–(6) are equivalent, and if moreover X is a smooth variety, then (7) is also equivalent.

  1. 1. w is a $\mu _{p^e}$ -fixed point.

  2. 2. The maximal ideal $\mathfrak {m}_w$ of $\mathcal {O}_{X,w}$ is generated by homogeneous elements.

  3. 3. The canonical morphism $B \to B / \mathfrak {m}_w$ is $\mu _{p^e}$ -equivariant, where $B/\mathfrak {m}_w$ is equipped with the trivial action (i.e., the decomposition concentrated on $(-)_0$ ).

  4. 4. $B_i \subset \mathfrak {m}_w$ for each $i \neq 0$ .

  5. 5. $D(\mathfrak {m}_w) \subset \mathfrak {m}_w$ .

  6. 6. $D(\mathcal {O}_{X,w}) \subset \mathfrak {m}_w$ .

  7. 7. D has singularity at w in the sense of [Reference OgusRS, §1].

Finally, if (1) holds, then the $\mu _{p^e}$ -action extends to the blowup $\operatorname {\mathrm {Bl}}_w X$ .

Proof. Let $B = \bigoplus _{i \in \mathbb Z/p^e\mathbb Z} B_i$ be the corresponding decomposition.

(1 $\iff $ 3) By the definition of $X^{\mu _p}$ , a closed point $w \in X$ is a (k-valued) point of $X^{\mu _p}$ if and only if $B \to B/\mathfrak {m}_w$ is compatible with the projections $\operatorname {\mathrm {pr}}_i$ to the ith summand $(-)_i$ for all i, where $B/\mathfrak {m}_w$ is equipped with the trivial decomposition.

(2 $\iff $ 3) If (3) holds, then we have $\operatorname {\mathrm {pr}}_i(\mathfrak {m}_w) \subset \mathfrak {m}_w$ for all i, and then each element x of $\mathfrak {m}_w$ is the sum of homogeneous elements $\operatorname {\mathrm {pr}}_i(x) \in \mathfrak {m}_w$ . Conversely, if $\mathfrak {m}_w$ is generated by homogeneous elements, then $\operatorname {\mathrm {pr}}_i(\mathfrak {m}_w) \subset \mathfrak {m}_w$ for all i, which implies (3).

(3 $\iff $ 4) Easy.

Assume $e = 1$ .

(2 $\iff $ 5) Assume $D(\mathfrak {m}_w) \subset \mathfrak {m}_w$ . Take a system of generators $(x_j)$ of $\mathfrak {m}_w$ . For each j, let $x_j = \sum _{i \in \mathbb F_p} x_{j,i}$ be the decomposition of $x_j$ in $B = \bigoplus _{i} B_i$ . Then $D^l(x_j) = \sum _i i^l x_{j,i}$ is also in $\mathfrak {m}_w$ . Since the matrix $(i^l)_{i,l=0}^{p-1}$ is invertible, this implies that $x_{j,i} \in \mathfrak {m}_w$ . Thus $\mathfrak {m}_w$ is generated by eigenvectors. The converse is clear.

(5 $\iff $ 6) This is clear since $\mathcal {O}_{X,w} = \mathfrak {m}_w + k$ and $D \rvert _{k} = 0$ .

(5 $\iff $ 7) Take coordinates $x_1, \ldots , x_n$ at a point w and write $D = \sum _j f_j \cdot (\partial / \partial x_j)$ . Then both of the conditions are equivalent to $(f_j) \subset \mathfrak {m}_w$ .

We show the final assertion assuming (2). If the maximal ideal $\mathfrak {m}$ is generated by homogeneous elements $x_j \in B_{i_j}$ , then, for each j, we can extend the action on the affine piece $\operatorname {\mathrm {Spec}} B[x_h/x_j]_h$ of $\operatorname {\mathrm {Bl}}_w X$ by declaring $x_h/x_j$ to be homogeneous of weight $i_h - i_j$ .

The next lemma enables us to take useful coordinates at a point not fixed by D.

Lemma 2.9. If B is a Noetherian local ring, D is a derivation of multiplicative type, and the closed point is not fixed by D, then the maximal ideal $\mathfrak {m}$ of B is generated by elements $x_1, \ldots , x_{m-1}, y$ with $\operatorname {\mathrm {wt}}(x_j) = 0$ and $\operatorname {\mathrm {wt}}(1+y) = 1$ . If $\mathfrak {m}$ is generated by n elements, then we can take $m = n$ . If $\dim B \geq 2$ , then D does not extend to a derivation of the blowup $\operatorname {\mathrm {Bl}}_{\mathfrak {m}} B$ .

Proof. Recall that a subset of $\mathfrak {m}$ generates $\mathfrak {m}$ if and only if it generates $\mathfrak {m}/\mathfrak {m}^2$ .

Take elements $x^{\prime }_1, \dots , x^{\prime }_m$ generating $\mathfrak {m}$ , and let $x^{\prime }_j = \sum _{i \in \mathbb F_p} x^{\prime }_{j,i}$ be the decompositions to eigenvectors. By assumption, there exists a pair $(j,i)$ with $x^{\prime }_{j,i} \not \in \mathfrak {m}$ . We take such $j_0,i_0$ , and we may assume $i_0 \neq 0$ . We may assume $x^{\prime }_{j_0,i_0} - 1 \in \mathfrak {m}$ . Then $y = x^{\prime }_{j_0,i_0} - 1 $ satisfies $y \in \mathfrak {m}$ and $D(y) = i_0 (y + 1)$ . We have $y \not \in \mathfrak {m}^2$ , since $D(\mathfrak {m}^2) \subset \mathfrak {m}$ . By replacing y with $(y+1)^q - 1$ for an integer q with $q i_0 \equiv 1 \pmod p$ , we may assume $i_0 = 1$ . For each j, let $x_j = \sum _i (y+1)^{-i} x^{\prime }_{j,i}$ . Then we have $D(x_j) = 0$ and, since $x_j \equiv x^{\prime }_j \pmod {(y)}$ , the elements $x_j,y$ generate $\mathfrak {m}/\mathfrak {m}^2$ and hence generate $\mathfrak {m}$ . We can omit one of the $x_j$ ’s and then the remaining elements satisfies the required conditions (after renumbering).

To show the latter assertion, it suffices to show that D does not extend to $B' := B[x_j/y]_j$ . If it extends, then we have $D(x_j/y) = -x_j(y+1)/y^2 \in B'$ , hence $x_j/y^2 \in B'$ , and then on $\operatorname {\mathrm {Spec}} B'$ we have that $y = 0$ implies $x_j/y = 0$ , which is impossible since $\dim B' \geq 2$ .

Before stating the next proposition, we recall the following notion from [Reference OgusRS]. Assume X is a smooth irreducible variety and D is a nontrivial derivation. Then $\operatorname {\mathrm {Fix}}(D)$ consists of its divisorial part $(D)$ and non-divisorial part $\langle D \rangle $ . If we write $D = f \sum _i g_i \frac {\partial }{\partial x_i}$ for some local coordinates $x_1, \dots , x_m$ with $g_i$ having no common factor, then $(D)$ and $\langle D \rangle $ correspond to the ideals $(f)$ and $(g_i)$ , respectively. If D is of multiplicative type with $\langle D \rangle = \emptyset $ , then it follows from Proposition 2.8 that for suitable coordinates near any fixed point, we have $D = a x_m \cdot (\partial / \partial x_m)$ and that $\operatorname {\mathrm {Fix}}(D)$ is a smooth divisor (possibly empty).

Assuming that $\operatorname {\mathrm {Fix}}(D)$ is divisorial, in which case the quotient is a smooth variety by [Reference Rudakov and ShafarevichS, Prop. 6], the highest differential forms on smooth loci of X and $X^D$ are related in the following way.

Proposition 2.10. Let X be a smooth variety of dimension m (not necessarily proper) equipped with a nontrivial derivation D of multiplicative type such that $\operatorname {\mathrm {Fix}}(D)$ is divisorial. Let $\Delta $ be the divisor $\operatorname {\mathrm {Fix}}(D)$ . Then there is a unique collection of isomorphisms

$$\begin{align*}(\pi_* (\Omega^m_{X/k}(\Delta))^{\otimes n})_0 \cong (\Omega^m_{X^D/k}(\pi_*(\Delta)))^{\otimes n} \end{align*}$$

for all integers n, compatible with multiplication, preserving the zero loci, and sending (for $n = 1$ )

$$\begin{align*}f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m) \mapsto f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m^p) \end{align*}$$

if $f_0, \dots , f_{m-1}$ are homogeneous of weight $0$ and $f_m$ is homogeneous of some weight (not necessarily $0$ ).

In particular, if the action is fixed-point-free, then we have isomorphisms

$$ \begin{align*} (\pi_* (\Omega^m_{X/k})^{\otimes n})_0 &\cong (\Omega^m_{X^D/k})^{\otimes n} \quad \text{and} \\ H^0(X, (\Omega^m_{X/k})^{\otimes n})_0 &\cong H^0(X^D, (\Omega^m_{X^D/k})^{\otimes n}) \end{align*} $$

with the same properties.

Proof. The isomorphism for $n = 0$ is clear. It suffices to construct the isomorphism for $n = 1$ that is compatible with multiplication with $n = 0$ forms and with restriction to open subschemes.

Take a closed point $w \in X$ . Let $\varepsilon = 1$ (resp. $\varepsilon = 0$ ) if $w \not \in \Delta $ (resp. $w \in \Delta $ ). By Lemma 2.9 (resp. by [Reference OgusRS, Th. 1]), there are coordinates $x_1, \dots , x_m$ on a neighborhood of w with $D(x_j) = 0$ for $j < m$ and $D(x_m) = a (\varepsilon + x_m)$ for some $a \in \mathbb F_p^*$ . We define

$$ \begin{align*} \phi \colon (\pi_* (\Omega^m_{X/k}(\Delta)))_0 &\to \Omega^m_{X^D/k}(\pi_*(\Delta)) \\ f \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m) &\mapsto f \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m^p) \end{align*} $$

for f of weight $0$ (note that $dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log (\varepsilon + x_m)$ is a local generator of the left-hand side). We show that then $\phi $ sends

$$\begin{align*}f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m) \mapsto f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m^p) \end{align*}$$

for any $f_0, \dots , f_{m-1}$ and $f_m$ as in the statement. This implies that $\phi $ does not depend on the choice of the coordinates and hence that $\phi $ induces a well-defined morphism of sheaves. Then since $dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log (\varepsilon + x_m)$ (resp. $dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log (\varepsilon + x_m^p)$ ) is a local generator of $(\Omega ^m_{X/k}(\Delta ))_0$ (resp. $\Omega ^m_{X^D/k}(\pi _*(\Delta ))$ ), it follows that $\phi $ is an isomorphism and $\phi ^{\otimes n}$ are well-defined isomorphisms.

We may pass to the completion, so consider $f_h \in k[[x_1, \dots , x_m]]$ . By the assumption on the weight, we have $f_h \in k[[x_1, \dots , x_{m-1}, x_m^p]]$ for $h < m$ and $f_m \in (\varepsilon + x_m)^b k[[x_1, \dots , x_{m-1}, x_m^p]]$ for some $0 \leq b < p$ . Then we have $\partial f_h / \partial x_m = 0$ for $h < m$ and $\partial f_m / \partial x_m = b f_m/(\varepsilon + x_m)$ . Hence, we have

$$ \begin{align*} & f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m) \\ &= f_0 ((\varepsilon + x_m)/f_m) \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m) \\ &= b f_0 \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m-1} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m) \\ &\stackrel{\phi}{\mapsto} b f_0 \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m-1} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m^p). \end{align*} $$

On the other hand, in the invariant subalgebra $k[[x_1, \dots , x_{m-1}, x_m^p]]$ , we have $\partial f_m^p / \partial x_j = 0$ for $j < m$ and $\partial f_m^p / \partial x_m^p = b f_m^p/(\varepsilon + x_m^p)$ . Hence, we have

$$ \begin{align*} & f_0 \cdot df_1 \wedge \dots \wedge df_{m-1} \wedge d\log(f_m^p) \\ &= \dots = b f_0 \det (\partial f_h / \partial x_j)_{1 \leq h,j \leq m-1} \cdot dx_1 \wedge \dots \wedge dx_{m-1} \wedge d\log(\varepsilon + x_m^p). \end{align*} $$

The assertion follows.

We will give another abstract proof of Proposition 2.10 in [Reference MatsumotoMat3, Prop. 2.12].

2.4 Global properties of derivations

Lemma 2.11. Let $C \subset \mathbb P^2$ be a quadratic curve (not necessarily irreducible nor reduced) in characteristic p, and let D be a p-closed derivation. Then $\operatorname {\mathrm {Fix}}(D) \neq \emptyset $ .

Proof. Suppose C is integral. Then $C \cong \mathbb P^1$ and the result is well known (indeed, $T_{\mathbb P^1} \cong \mathcal {O}_{\mathbb P^1}(2)$ ).

Suppose C is reducible. We may assume $C = (xy = 0)$ . Let $U = \operatorname {\mathrm {Spec}} k[x,y] / (xy)$ . Then $T_U = ( x \frac {d}{dx} - y \frac {d}{dy}) \cdot \mathcal {O}_U$ and hence the origin belongs to $\operatorname {\mathrm {Fix}}(D)$ .

Suppose C is non-reduced. We may assume $C = (X_3^2 = 0)$ . If $p \neq 2$ , then D induces a derivation $D_{\mathrm {red}}$ on $C_{\mathrm {red}} \cong \mathbb P^1$ , and we have $\operatorname {\mathrm {Fix}}(D) \approx \operatorname {\mathrm {Fix}}(D_{\mathrm {red}}) \neq \emptyset $ . Suppose $p = 2$ . It is easy to see that $H^0(C, T_C) \stackrel {\sim }{\to } H^0(C, T_{\mathbb P^2} \rvert _{C}) \stackrel {\sim }{\leftarrow } H^0(\mathbb P^2, T_{\mathbb P^2})$ . Hence, there exist $f_1, f_2, f_3 \in H^0(\mathbb P^2, \mathcal {O}(1)) = \bigoplus _{i=1}^3 k X_i$ such that $D\bigl (\frac {X_i}{X_j}\bigr ) = \frac {f_i}{X_j} - \frac {X_i f_j}{X_j^2}$ . If $f_3 \in k X_3$ , then D induces a derivation $D_{\mathrm {red}}$ on $C_{\mathrm {red}} \cong \mathbb P^1$ , and we conclude as above. Suppose $f_3 \not \in k X_3$ . By a coordinate change, we may assume $f_3 - X_2 \in k X_3$ . Letting $x_i = X_i/X_1$ ( $i = 2,3$ ) and restricting to $\operatorname {\mathrm {Spec}} k[x_2,x_3]/(x_3^2) = (X_1 \neq 0) \subset C$ , we have $D(x_3) - x_2 \in (x_3)$ , in particular $D(x_3) \in \mathfrak {m} := (x_2, x_3)$ . If $D(x_2) \in \mathfrak {m}$ , then the origin is a fixed point. Suppose $D(x_2) \not \in \mathfrak {m}$ and $D^2 = h D$ . Then $h = D^2(x_2) / D(x_2) \in \mathcal {O}_{\mathfrak {m}}$ , and hence $D(x_2) \equiv D^2(x_3) = h D(x_3) \equiv 0 \pmod {\mathfrak {m}}$ , contradiction.

Corollary 2.12. Suppose $\mu _p$ acts on an RDP surface X and fixes an RDP w. Then the action extends to the blowup $\operatorname {\mathrm {Bl}}_w X$ and there exists a fixed point above w.

Proof. The action extends to the blowup by Proposition 2.8. Let $D'$ be the induced derivation on $\operatorname {\mathrm {Bl}}_w X$ . Let $C \subset \operatorname {\mathrm {Bl}}_w X$ be the (possibly non-reduced) exceptional divisor, which is a quadratic curve in $\mathbb P^2$ since w is an RDP. Since $D'(\mathcal I_C) \subset \mathcal I_C$ , $D'$ induces a derivation $D^{\prime }_{C}$ (of multiplicative type) on C. By Lemma 2.11, $D^{\prime }_C$ has at least one fixed point, and that point is also a fixed point of $D'$ .

Later we will also need the following Katsura–Takeda formula on rational vector fields (i.e., derivations on the fraction field $k(X)$ ). For a rational derivation D locally of the form $f^{-1} D'$ for some regular function f and (regular) derivation $D'$ , we define the divisorial and nondivisorial parts by $(D) = (D') - \operatorname {\mathrm {div}}(f)$ and $\langle D \rangle = \langle D' \rangle $ .

Proposition 2.13 [Reference KatsuraKT, Prop. 2.1]

Let X be a smooth proper surface, and let D be a nonzero rational vector field. Then we have

$$\begin{align*}\deg c_2(X) = \deg \langle D \rangle - K_X \cdot (D) - (D)^2. \end{align*}$$

3 Tame symplectic actions on RDPs

Hereafter, all action of groups and group schemes on schemes are assumed faithful.

Throughout this section, we work under the following setting. $B = \mathcal {O}_{X,w}$ is the localization of an RDP surface X over an algebraically closed field k at a closed point w (either a smooth point or an RDP), $\mathfrak {m} \subset B$ is the maximal ideal, G is a finite group acting on X, and the action restricts to $\operatorname {\mathrm {Spec}} B$ . Assume that the order of G is not divisible by $p = \operatorname {\mathrm {char}} k$ .

Definition 3.1. We say that the G-action on B is symplectic if it acts on the one-dimensional k-vector space $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k}) \otimes _B (B/\mathfrak {m})$ trivially.

If $G = \mathbb Z/p\mathbb Z$ , then any action is symplectic (cf. Remark 2.6), and hence the notion is useless in this case.

Remark 3.2. If B is as above and the G-action is symplectic, then the rank- $1$ free B-module $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ admits a generator $\omega $ that is G-invariant. Indeed, take a generator $\omega '$ , then $\omega := (1 / \lvert G \rvert ) \sum _{g \in G} g^* \omega '$ is clearly G-invariant and it is nonvanishing, since it is nonvanishing after $\otimes (B / \mathfrak {m})$ .

Remark 3.3. If X is an RDP K3 surface and $w \in X$ is a fixed closed point, then this is consistent with the usual notion of symplecticness, since a generator of $H^0(X^{\mathrm {sm}}, \Omega ^2) \cong H^0(\tilde {X}, \Omega ^2)$ (Proposition 2.3) restricts to a generator of this one-dimensional space. Thus, the symplecticness of an automorphism of an RDP K3 surface can be checked locally at any fixed point (if there exists any). The same for abelian surfaces.

Proposition 3.4. Let B and G be as above (in particular, the order of G is not divisible by $p = \operatorname {\mathrm {char}} k$ ). Then the invariant ring $B^G$ is again the localization at a closed point of an RDP surface.

Let $\tilde {X} \to X$ be the minimal resolution at w. Then $\tilde {X}/G \to X/G$ is crepant.

Proof. Let $\omega $ be a generator of the rank- $1$ free B-module $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ . By Remark 3.2, we may assume $\omega $ is G-invariant. The action of G on X induces an action on $\tilde {X}$ , and $\omega $ extends to a regular nonvanishing $2$ -form on $\tilde {X}$ . At each closed point $w' \in \tilde {X}$ , the stabilizer $G_{w'} \subset G$ acts on $T_{w'} \tilde {X}$ via $\operatorname {\mathrm {SL}}_2(k)$ since G preserves $\omega $ . Hence, the quotient $\tilde {X}/G$ has only RDPs as singularities. Since $\omega $ is preserved by G, it induces a regular nonvanishing $2$ -form on $(\tilde {X}/G)^{\mathrm {sm}}$ , and since RDPs are canonical singularities, it extends to a regular nonvanishing $2$ -form on $\widetilde {\tilde {X}/G}$ , the minimal resolution of $\tilde {X}/G$ above w. Thus, $B^G$ is a canonical singularity, that is, either a smooth point or an RDP.

Remark 3.5. We [Reference Madapusi PeraMat1, Prop. 3.8] described possible symplectic actions of finite tame groups on RDPs. For actions of cyclic groups $G = \mathbb Z/n\mathbb Z$ ( $n> 1$ ), we have a complete classification: possible n and the types of X and $X/G$ are listed in Table 2.

Table 2 Tame symplectic cyclic actions on RDPs.

Remark 3.6. Singularities of quotients by order p automorphisms in characteristic $p> 0$ tends to be worse than those in characteristic $\neq p$ . For example, the quotient of a supersingular abelian surface in characteristic $2$ by the automorphism $x \mapsto -x$ is a rational surface with an elliptic singularity [Reference JangKa, Th. C].

4 $\mu _n$ -actions on RDPs and quotients

Throughout this section, we work under the following setting. $B = \mathcal {O}_{X,w}$ is the localization of an RDP surface X over an algebraically closed field k of characteristic $p \geq 0$ at a closed point w (either a smooth point or an RDP), $\mathfrak {m} \subset B$ is the maximal ideal, n is a positive integer possibly divisible by p, $\mu _n$ acts on X, and the action restricts to $\operatorname {\mathrm {Spec}} B$ . (Note that w is not necessarily fixed by $\mu _n$ .)

If $n = p> 0$ , then the corresponding derivation of multiplicative type is denoted by D.

4.1 Symplecticness of $\mu _n$ -actions

Assume w is fixed by the $\mu _n$ -action. Then the action on B induces an action on $V := H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k}) \otimes _B (B/\mathfrak {m})$ , that is, a decomposition $V = \bigoplus _{i \in \mathbb Z/n\mathbb Z} V_i$ of k-vector spaces. Since $\dim _k V = 1$ , V is equal to one of the summands. In other words, V is of some weight $i_0 \in \mathbb Z/n\mathbb Z$ .

Definition 4.1. We say that the $\mu _{n}$ -action, or the corresponding derivation if $n = p$ , on B is symplectic if V is of weight $0$ .

We say that a $\mu _{n}$ -action, or a derivation D of multiplicative type, on an RDP surface X is symplectic at a fixed closed point w if the induced action or derivation on $\mathcal {O}_{X,w}$ is symplectic in the above sense.

Remark 4.2. If $p \nmid n$ , then $\mu _n$ is (noncanonically) isomorphic to $\mathbb Z/n\mathbb Z$ and this definition is consistent with Definition 3.1.

Remark 4.3 (cf. Remark 3.2)

If B is as above and V is of weight $i_0$ , then the rank- $1$ free B-module $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ admits a generator $\omega $ of weight $i_0$ . Indeed, take a generator $\omega '$ , let $\omega ' = \sum _{i} \omega ^{\prime }_i$ be its decomposition, and write $\omega ^{\prime }_i = f_i \omega '$ with $f_i \in B$ . Since $\sum f_i = 1$ , there exists $i_1 \in \mathbb Z/n\mathbb Z$ with $f_{i_1} \in B^*$ . Then $i_0 = i_1$ and hence we can take $\omega = \omega ^{\prime }_{i_1}$ . If $n = p$ , then this means $D(\omega ) = i_0 \omega $ .

From this, it follows that if $\mu _{n}$ acts on an RDP surface, then the weight $i_0$ is a locally constant function on the fixed locus.

Remark 4.4 (cf. Remark 3.3)

If X is an RDP K3 surface and $w \in X$ is a fixed closed point, then the action is symplectic in the sense of Definition 2.5 if and only if action is symplectic at w, since a generator of $H^0(X^{\mathrm {sm}}, \Omega ^2)$ restricts to a generator of this one-dimensional space. Thus, the symplecticness of a $\mu _{n}$ -action on an RDP K3 surface can be checked locally at any fixed point (if there exists any). The same for abelian surfaces.

Lemma 4.5. Suppose the closed point w of B is fixed under the $\mu _n$ -action. Then B is generated by $2$ or $3$ homogeneous elements, respectively, if B is smooth or an RDP. Moreover:

  1. 1. If B is smooth and generated by elements $x,y$ of respective weights $a,b$ , then the action is symplectic if and only if $a + b = 0$ (in $\mathbb Z/n\mathbb Z$ ).

  2. 2. If B is an RDP and generated by $x,y,z$ of respective weights $a,b,c$ , then there is $d \in \mathbb Z/n\mathbb Z$ and a homogeneous power series $F \in k[[x,y,z]]$ of weight d such that $\hat {B} \cong k[[x,y,z]]/(F)$ . The action is symplectic if and only if $a + b + c = d$ .

Proof. The first assertion follows from Proposition 2.8.

(1) $H^0(\operatorname {\mathrm {Spec}} B \setminus \{\mathfrak {m}\}, \Omega ^2_{B/k})$ is generated by $dx \wedge dy$ , which is of weight $a+b$ .

(2) Take an element $H \in k[[x,y,z]]$ such that $\hat {B} = k[[x,y,z]] / (H)$ , and let $H = \sum _{i \in \mathbb Z/n\mathbb Z} H_i$ be the decomposition with respect to the $\mu _n$ -action. Since $H = 0$ in $\hat {B}$ , we have $H_i = 0$ in $\hat {B}$ , and hence there are $f_i \in k[[x,y,z]]$ such that $H_i = f_i H$ . Since $\sum f_i = 1$ , there exists $d \in \mathbb Z/n\mathbb Z$ with $f_d \in k[[x,y,z]]^*$ . We can take $F = H_d$ , which is of weight d.

Then $H^0(\operatorname {\mathrm {Spec}} \hat {B} \setminus \{\mathfrak {m}\}, \Omega ^2_{\hat {B}/k})$ is generated by $\omega = F_x^{-1} dy \wedge dz = F_y^{-1} dz \wedge dx = F_z^{-1} dx \wedge dy$ (this means that the restriction of $\omega $ to the open subscheme $\operatorname {\mathrm {Spec}} \hat {B}[F_x^{-1}]$ is equal to $F_x^{-1} dy \wedge dz$ , and so on), and we have $\operatorname {\mathrm {wt}}(\omega ) = a + b + c - d$ since $\operatorname {\mathrm {wt}}(F_x^{-1}) = -(d - a)$ and $\operatorname {\mathrm {wt}}(dy \wedge dz) = b + c$ , and so on.

4.2 $\mu _p$ -actions on RDPs

As noted in §2.3, we know by [Reference Rudakov and ShafarevichS, Prop. 6] (see also [Reference OgusRS, Th. 1 and Corollary]) that the quotient of a smooth variety by a $\mu _p$ -action with no isolated fixed point is smooth. We need to consider, more generally, the quotients of surfaces with RDP singularities and with isolated fixed points.

Let $\mathcal {O}_{X,w}$ and $\mu _n$ be as in the beginning of §4, and suppose $n = p$ . Let $\pi \colon X \to Y = X/\mu _p$ be the quotient morphism.

Theorem 4.6.

  1. 1. Assume w is non-fixed. If w is a smooth point, then $\pi (w) \in Y$ is also a smooth point. If w is an RDP, then $\pi (w)$ is either a smooth point or an RDP. In either case, $X \times _Y \tilde {Y} \to X$ is crepant, where $\tilde {Y} \to Y$ is the minimal resolution at $\pi (w)$ .

  2. 2. If w is fixed and the action is symplectic at w, then w is an isolated fixed point and $\pi (w)$ is an RDP.

  3. 3. If w is an isolated fixed point and the action is non-symplectic at w, then $\pi (w)$ is a non-RDP singularity.

First, we consider non-symplectic actions on isolated fixed points.

Proof of Theorem 4.6(3)

By Proposition 2.10, we have an isomorphism

$$\begin{align*}(H^0(\operatorname{\mathrm{Spec}} \mathcal{O}_{X,w} \setminus \{w\}, \Omega^2))_0 \cong H^0(\operatorname{\mathrm{Spec}} \mathcal{O}_{Y,\pi(w)} \setminus \{\pi(w)\}, \Omega^2) \end{align*}$$

preserving the zero loci of $2$ -forms. If $\pi (w)$ is either a smooth point or an RDP, then the right-hand side has a nonvanishing $2$ -form and hence there is a nonvanishing form $\omega $ on $\operatorname {\mathrm {Spec}} \mathcal {O}_{X,w} \setminus \{w\}$ of weight $0$ . Being nonvanishing, $\omega $ is a generator of $H^0(\operatorname {\mathrm {Spec}} \mathcal {O}_{X,w} \setminus \{w\}, \Omega ^2)$ . However, this contradicts the non-symplecticness assumption.

Next, we consider non-fixed points. In fact, we can classify all possible actions and give explicit equations.

Proposition 4.7. Assume w is not fixed.

  • If w is a smooth point, then there are coordinates $x,y$ of $\mathcal {O}_{X,w}$ satisfying $D(x) = 0$ and $D(y) \neq 0$ , and hence $\mathcal {O}_{Y,\pi (w)}$ has $x,y^p$ as coordinates and in particular $\pi (w)$ is a smooth point.

  • If w is an RDP, then there is an element $F \in k[[x,y,z^p]]$ and an isomorphism $\hat {\mathcal {O}}_{X,w} \cong k[[x,y,z]] / (F)$ with $D(x) = D(y) = 0$ and $D(z) \neq 0$ , and hence $\hat {\mathcal {O}}_{Y,\pi (w)} \cong k[[x,y,z^p]] / (F)$ . Moreover, we can take F to be one in Table 3.

Table 3 Non-fixed $\mu _p$ -actions on RDPs.

Proof of Theorem 4.6(1) and Proposition 4.7

If w is a smooth point, then taking coordinates $x,y$ as in Lemma 2.9 (i.e., $D(x) = 0$ and $D(y) = 1+y$ ), we have $\hat {\mathcal {O}}_{Y,\pi (w)} \cong k[[x,y^p]]$ , and hence $\mathcal {O}_{Y,\pi (w)}$ is smooth.

Assume w is an RDP. By Lemma 2.9, we have coordinates $x,y,z$ satisfying $D(x) = D(y) = 0$ and $D(z) \neq 0$ . We have $\hat {\mathcal {O}}_{X,w} \cong k[[x,y,z]] / (F)$ for some $F \in k[[x,y,z]]$ such that $D(F) \in (F)$ , and we may assume $F \in k[[x,y,z^p]]$ . We show that, after replacing F with a multiple by a unit, and after a coordinate change of $k[[x,y,z]]$ that preserves the subring $k[[x,y,z^p]]$ , F coincides with one in Table 3. (Such coordinate changes are given by $x',y',z' \in \mathfrak {m}$ that are linearly independent in $\mathfrak {m}/\mathfrak {m}^2$ and satisfy $x',y' \in \mathfrak {m} \cap k[[x,y,z^p]]$ .) A similar classification is given in [Reference Ekedahl, Hyland and Shepherd-BarronEH+, Prop. 3.8], but they missed the case of $E_7^0$ in characteristic $3$ .

Assume the classification for the moment. Then, in each case, we observe that $\pi (w)$ is either a smooth point or an RDP, and it is straightforward to check that $X \times _Y \tilde {Y}$ is an RDP surface crepant over X. (In Table 3, the entries of the singularities of $X \times _Y \tilde {Y}$ are omitted if Y is already smooth.) For example, consider $X = \operatorname {\mathrm {Spec}} k[x,y,z]/(F)$ , $F = xy + z^{mp}$ with $m \geq 2$ . Then $X' := X \times _Y \operatorname {\mathrm {Bl}}_{\pi (w)} Y$ is covered by three affine pieces

$$ \begin{align*} X^{\prime}_1 &= \operatorname{\mathrm{Spec}} k[x, y_1, v_1, z] / (y_1 + x^{m-2} v_1^{m}, x v_1 - z^p), & y_1 &= y/x, & v_1 &= z^p/x, \\ X^{\prime}_2 &= \operatorname{\mathrm{Spec}} k[x_2, y, v_2, z] / (x_2 + y^{m-2} v_2^{m}, y v_2 - z^p), & x_2 &= x/y, & v_2 &= z^p/y, \\ X^{\prime}_3 &= \operatorname{\mathrm{Spec}} k[x_3, y_3, z] / (x_3 y_3 + z^{(m-2)p}), & x_3 &= x/z^p, & y_3 &= y/z^p. \end{align*} $$

One observes that $\operatorname {\mathrm {Sing}}(X')$ consists of two RDPs of type $A_{p-1}$ at the origins of $X^{\prime }_1$ and $X^{\prime }_2$ and, if $m \geq 3$ , one RDP of type $A_{(m-2)p-1}$ at the origin of $X^{\prime }_3$ . Repeating this, we observe that $X \times _Y \tilde {Y}$ has $m A_{p-1}$ .

Now, we show the classification. We say that F has a monomial if the coefficient of that monomial is nonzero. We also write $F = \sum _{h,i,j} a_{hij} x^h y^i z^j$ .

First, assume $p> 2$ . We may assume that the degree $2$ part $F_2$ is either $xy$ or $x^2$ . Assume $F_2 = xy$ . We may assume that F has no $xz^{j}$ and $yz^{j}$ . F must have $z^{j}$ , $j = mp$ , and then it is $A_{mp-1}$ . Then, by replacing x with $x + a_{0ij} y^{i-1} z^j$ and y with $y + a_{h0j} x^{h-1} z^j$ , and so on, we may assume that F has no $y^i z^j$ with $i> 0$ and no $x^h z^j$ with $h> 0$ . Thus, $F = u_1 xy + u_2 z^{mp}$ for some units $u_1, u_2$ , and then by replacing $x,y,F$ by suitable multiples, we obtain $F = xy + z^{mp}$ .

Assume $p> 3$ and $F_2 = x^2$ . We may assume that the degree $3$ part $F_3$ is $y^3$ . If $p \geq 7$ , it cannot be an RDP. If $p = 5$ , then F must have $z^5$ , and then it is $E_8^0$ . We have $F = u_1 x^2 + u_2 y^3 + u_3 z^5$ , and then by replacing $x,y,F$ by suitable multiples, we obtain $F = x^2 + y^3 + z^5$ . (For example, we let $F = u_3 F'$ , $x = (u_3 u_1^{-1})^{1/2} x'$ , $y = (u_3 u_2^{-1})^{1/3} y'$ . Note that we can take nth roots of units provided $p \nmid n$ .)

Assume $p = 3$ and $F_2 = x^2$ . We may assume $F_3 = y^3$ or $F_3 = z^3$ . If $F_3 = z^3$ , then F must have $y^4$ or $y^5$ , and then it is $E_6^0$ or $E_8^0$ . We may assume $a_{130} = a_{140} = 0$ by replacing x with $x + (1/2)(a_{130} y^3 + a_{140} y^4)$ , and then we transform F as above. If $F_3 = y^3$ , then F must have $yz^3$ and then it is $E_7^0$ . We eliminate $a_{1ij}$ as above, then we have $F = u_1 x^2 + u_2 y^3 + u_3 yz^3 + z^6 g(z^3)$ for some power series $g \in k[[z^3]]$ . We may assume $u_i \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate g by replacing y with $y + z^3 g$ , and then we transform F as above.

Now, consider $p = 2$ . We may assume $F_2$ is one of $xy + z^2$ (if irreducible), $xy$ (if reducible but not a square), $z^2$ , or $x^2$ (square, of a linear factor containing z or not). If $F_2 = xy + z^2$ or $F_2 = xy$ , then as above, it is $A_{mp-1}$ and F becomes $xy + z^{mp}$ .

Assume $p = 2$ and $F_2 = x^2$ . If $F_3$ has $yz^2$ , then F must have $xy^m$ and then it is $D_{2m+1}^0$ . We obtain $F = u_1 x^2 + u_2 y z^2 + u_3 x y^m + z^4 g(z) + f(y) + y^{2m} g'(y)$ , where $f(y) = f_0(y)^2 + y f_1(y)^2$ is a polynomial of degree $< 2m$ , $g(z) \in k[[z]]$ , and $g'(y) \in k[[y]]$ . We may assume $u_i \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate f by replacing x with $x + f_0(y)$ and z with $z + f_1(y)$ and so on. Then we eliminate g and $g'$ by replacing y and x suitably, and take multiples by units as above.

If $F_3$ has no $yz^2$ , then F must have $y^3$ and $xz^2$ and then it is $E_6^0$ . We obtain $F = u_1 x^2 + u_2 y^3 + u_3 x z^2 + a y^2 z^2 + z^4 g$ , $g = g_0(z^2) + y g_1(z^2) + y^2 g_2(z^2)$ . We may assume $u_i \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate a and g by replacing y and x suitably, and then we transform F as above.

Assume $p = 2$ and $F_2 = z^2$ . Let $\overline {F_3} = (F_3 \bmod (z)) \in k[[x,y]]$ . If $\overline {F_3}$ has three distinct roots, then we may assume $\overline {F_3} = x^3 + y^3$ and then it is $D_4^0$ . We can transform F to $z^2 + x^3 + y^3$ as above, and then to $z^2 + x^2 y + x y^2$ . If $\overline {F_3}$ has two distinct roots, then we may assume $\overline {F_3} = x^2y$ and F must have $xy^m$ and then it is $D_{2m}^0$ . We obtain $F = u_1 z^2 + u_2 x^2 y + u_3 x y^m + g(x) + f(y) + y^{2m-1} g'(y)$ , where $f(y) = f_0(y)^2 + y f_1(y)^2$ is a polynomial of degree $< 2m-1$ and $g \in k[[x]]$ and $g' \in k[[y]]$ . We argue as in the case of $D_{2m+1}^0$ . If $\overline {F_3}$ has one (triple) root, then we may assume $\overline {F_3} = x^3$ and F must have $xy^3$ or $y^5$ and then it is $E_7^0$ or $E_8^0$ . We transform F as above.

Next, we consider symplectic actions on fixed points.

Lemma 4.8. Assume w is fixed, and the action is symplectic at w.

  1. 1. Assume w is a smooth point. Then w is an isolated fixed point and $\pi (w)$ is an RDP of type $A_{p-1}$ . The eigenvalues of D on the cotangent space $\mathfrak {m}_w / \mathfrak {m}_w^2$ are of the form $a,-a$ for some $a \in \mathbb F_p^*$ .

  2. 2. Assume w is an RDP. Let ${f}:{X' = \operatorname {\mathrm {Bl}}_w X}\to {X}$ . Then $X'$ is an RDP surface, D uniquely extends to a derivation $D'$ on $X'$ which is symplectic at every fixed point above w, and $g \colon Y' = (X')^{D'} \to Y$ is crepant.

Proof. (1) By Lemma 4.5(1), we have $D = a x \cdot (\partial / \partial x) - a y \cdot (\partial / \partial y)$ with $a \in \mathbb F_p$ for some coordinates $x,y$ , and $a \neq 0$ since D is nontrivial. Hence, w is an isolated fixed point of D. We observe that $a,-a$ are the eigenvalues of the action on the cotangent space. We have $\hat {\mathcal {O}}_{X,w}^D = k[[x^p, xy, y^p]]$ , and it is an RDP of type $A_{p-1}$ .

(2) By Remark 4.3 and assertion (1), w is an isolated fixed point. By Proposition 2.8, D uniquely extends to $D'$ on $X'$ . Let $\omega $ be a generator of $H^0(\operatorname {\mathrm {Spec}} \mathcal {O}_{X,w} \setminus \{w\}, \Omega ^2)$ with $D(\omega ) = 0$ . Since w is an RDP, $X'$ is again an RDP surface, and it follows from Proposition 2.3 that $\omega $ extends to $\omega '$ on $(X')^{\mathrm {sm}}$ , which generates $H^0(\operatorname {\mathrm {Spec}} \mathcal {O}_{X',w'} \setminus \{w'\}, \Omega ^2)$ at any closed point $w' \in X'$ above w, and that $D'(\omega ') = 0$ . Hence, $D'$ is symplectic at every fixed point above w. Since as above such fixed points are isolated, $Y'$ is smooth outside finitely many isolated points. Applying Proposition 2.10 to $\omega $ on $X \setminus \{w\}$ and $\omega '$ on $X' \setminus (\operatorname {\mathrm {Sing}}(X') \cup \operatorname {\mathrm {Fix}}(D'))$ , we obtain $2$ -forms $\psi $ on $Y \setminus \{\pi (w)\}$ and $\psi '$ on $Y' \setminus \pi ((\operatorname {\mathrm {Sing}}(X') \cup \operatorname {\mathrm {Fix}}(D')))$ , which are nonvanishing. Comparing $\psi $ and $\psi '$ , we observe that g is crepant.

Proof of Theorem 4.6(2)

By Remark 4.3 and Lemma 4.8(1), w is an isolated fixed point. By shrinking X, we may assume that D has no fixed point except w.

We construct a finite sequence $(X_j,D_j)_{0 \leq j \leq n}$ ( $n \geq 0$ ) of RDP surfaces $X_j$ and derivations $D_j$ on $X_j$ of multiplicative type that is symplectic at each fixed point. Let $(X_0,D_0) = (X,D)$ . If $X_j$ has no fixed RDP, then we terminate the sequence at $n = j$ . If $X_j$ has at least one fixed RDP, let $X_{j+1}$ be the blowup of $X_j$ at the fixed RDPs and $D_{j+1}$ the extension of $D_j$ to $X_{j+1}$ . Since any RDP becomes smooth after a finite number of blowups at RDPs, this sequence terminates at some $n \geq 0$ . By Lemma 4.8(2), $D_{j+1}$ on $X_{j+1}$ is symplectic at each fixed point, and $(X_{j+1})^{D_{j+1}} \to (X_j)^{D_j}$ is crepant. By Theorem 4.6(1) and Lemma 4.8(1), $Y_n = (X_n)^{D_n}$ has canonical singularity (i.e., has no singularity other than RDPs), and since $Y_n \to Y = X^D$ is crepant, also Y has canonical singularity. If $n> 0$ , then $\pi (w)$ is not a smooth point since $Y_n \to Y$ is a crepant morphism non-isomorphic at that point, and if $n = 0$ , then $\pi (w)$ is not a smooth point by Lemma 4.8(1). Hence, in either case, $\pi (w)$ is an RDP.

Moreover, we can classify all possible symplectic $\mu _p$ -actions on RDPs.

Proposition 4.9. Assume w is a fixed RDP, and the action is symplectic at w. Then there is a $\mu _p$ -equivariant isomorphism $\hat {\mathcal {O}}_{X,w} \cong k[[x,y,z]]/(F)$ with F equal to one in Table 4 and $\mu _p$ acts on $x,y,z$ by respective weights $a,-a,0$ for some $a \in \mathbb F_p^*$ . The singularities of X, $X' = \operatorname {\mathrm {Bl}}_w X$ , $X/\mu _p$ , and $X'/\mu _p$ are displayed in the table.

Table 4 Symplectic $\mu _p$ -actions on RDPs.

$\bullet \ A_0$ is a smooth point that is an isolated fixed point of D.

$\bullet \ [\mathsf {n}]$ means that the RDP is not fixed by D.

$\bullet \ \lfloor q \rfloor := \max \{n \in \mathbb Z \mid n \leq q\}$ denotes the integer part of a real q.

$\bullet \ q^+ := \max \{q,0\}$ denotes the positive part of a real q.

$\bullet \ [*]$ : It follows from the classification that for each (formal) isomorphism class of RDP, there

exists only one fixed symplectic $\mu _p$ -action up to isomorphism, except for the case of $D_{2n}^{n-1}$ ( $n \geq 3$ )

in $p = 2$ , in which case there are two and they are distinguished by the degree $2$ part $F_2$ being a

square of a homogeneous element or not. We distinguish them by notation $D_{2n}^{n-1}$ and $D_{2n}^{n-1}[*]$ . We

use the convention that $D_4^{1}[*] = D_4^{1}$ .

Remark 4.10. A polynomial $f \in k[x_1, \dots , x_m]$ is called quasi-homogeneous if, for some $a_1, \dots , a_m \in \mathbb Z_{\geq 1}$ , the monomials appearing in f have the same degree with respect to a (i.e., degree of the monomial $x_1^{i_1} \dots x_m^{i_m}$ is $i_1 a_1 + \dots + i_m a_m$ ). RDPs whose completions are not defined by quasi-homogeneous polynomials, which exist only if $p = 2,3,5$ , are precisely $D_n^r$ and $E_n^r$ with $r \neq 0$ . It follows from the classification given in Proposition 4.9 (resp. given in Proposition 4.7, resp. which is omitted) that if an RDP of type $D_n$ or $E_n$ admits a fixed symplectic (resp. non-fixed, resp. fixed non-symplectic) $\mu _p$ -action, then the singularity is not defined (resp. is defined, resp. is defined) by a quasi-homogeneous polynomial. We do not know any explanation of this phenomenon.

Proof of Proposition 4.9

We consider tuples $(a,b,c,F)$ with $a,b,c \in \mathbb F_p$ , not all $0$ , and $F \in k[[x,y,z]]$ such that $F = 0$ defines an RDP and only monomials of weight $a+b+c$ ( $\in \mathbb F_p$ ) appear in F, where $x,y,z$ have respective weights $a,b,c$ . By Lemma 4.5(2), it suffices to consider $k[[x,y,z]]/(F)$ of this form. We show that there exist a $\mu _p$ -equivariant isomorphism $k[[x,y,z]]/(F) \cong k[[x',y',z']]/(F')$ with $F'$ in Table 4 and $\operatorname {\mathrm {wt}}(x',y',z') = (1,-1,0)$ up to $\operatorname {\mathrm {Aut}}(\mu _p) = \mathbb F_p^*$ (which amounts to replacing $(a,b,c)$ with $(ta,tb,tc)$ for some $t \in \mathbb F_p^*$ ).

We write $F = \sum _{h,i,j} a_{hij} x^h y^i z^j$ , and we say that a polynomial or a formal power series has a monomial if its coefficient is nonzero.

First, assume the degree $2$ part $F_2$ is a non-square. Then we may assume that $F_2$ contains a non-square monomial, say $xy$ . (Indeed, if this is not the case, then $p \neq 2$ and $F_2$ contains at least two square monomials, say $x^2$ and $y^2$ , then x and y has the same weight, and then after a linear coordinate change, we may assume $F_2$ contains $xy$ .) Then we have $c = 0$ . If $a + b \not \in \{0,a,b\}$ , then $F \in (x,y)^2$ , which implies that $F = 0$ is not an RDP. If $a + b = a \neq 0$ , then $F \in (x)$ , again not an RDP. The same if $a + b = b \neq 0$ . So we have $a + b = 0$ and hence $F \in k[[x^p,xy,y^p,z]]$ . Since F cannot belong to $(x,y)$ , there exists an integer m such that F has the monomial $z^{m}$ . Let m be the smallest such integer. We have $F = u_1 z^m + u_2 xy + g_1(x^p) + g_2(y^p)$ for some units $u_1,u_2 \in k[[x^p, x y, y^p, z]]^*$ and power series $g_1, g_2$ . We may assume $u_1,u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We eliminate $g_1, g_2$ by replacing x with $x + g_2/y$ and y with $y + g_1/x$ (and repeating this), and we obtain $F = u_1 z^m + u_2 xy$ . By replacing $x,y,z,F$ with suitable multiples, we obtain $F = z^m + xy$ .

Next, assume $p \geq 3$ and $F_2$ is square. We may assume $F_2 = z^2$ . We may assume $F_3 \not \equiv 0 \pmod {z}$ . If $F_3$ has $x^2y$ , then by $2c = 2a + b = a + b + c$ , we have $b = 0$ and $a = c$ , and hence $F \in (x,z)^2$ , which is absurd. Hence, we may assume $F_3$ has $y^3$ . By $2c = 3b = a+b+c$ , we have $(a,b,c) = (a,2a,3a)$ . If F does not have $x^3$ , then $F \in (z^2, x^3z, xyz, x^6, x^4y, x^2y^2, y^3)$ , and $F = 0$ cannot define an RDP. Hence, F has $x^3$ , hence $p = 3$ , and then $F \in k[[x^3,xy,y^3,z]]$ . We may assume that F does not have $xyz$ . To define an RDP, F must have one of $x^2y^2, x^4y, xy^4$ .

If it has $x^2y^2$ , then it is $E_6^1$ . We can eliminate $x^h y^i z$ , and we have

$$ \begin{align*} F &= z^2 u_1 + x^3 + y^3 + x^2 y^2 u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ &\qquad S_1 = \{(4,1,0), (1,4,0)\}, \qquad S_2 = \{(6,0), (7,1)\}, \qquad S_3 = \{(0,6), (1,7)\}, \end{align*} $$

where $a_{hij} \in k$ , $b_{hi} \in k[[x^3]]$ , $c_{hi} \in k[[y^3]]$ , and $u_1, u_2 \in k[[x^3, x y, y^3, z]]^*$ . By replacing x with $x + t y^2$ and y with $y + t' x^2$ , we eliminate $a_{410}$ and $a_{140}$ . Then, by replacing F with $(1 + x^3 b_{60} + x^4 y b_{71} + y^3 c_{06} + x y^4 c_{17}) F$ , we eliminate all $b_{hi}$ and $c_{hi}$ . Finally, we replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ . (For example, we let $F = u_2^{-3} F'$ , $x = u_2^{-1} x'$ , $y = u_2^{-1} y'$ , and $z = (u_1 u_2^3)^{-1/2} z'$ .)

If it does not have $x^2 y^2$ but has $x^4 y$ or $x y^4$ , then it is $E_8^1$ . By replacing F with a unit multiple, we may assume that it has $x^4 y$ and does not have $x y^4$ . We can eliminate $x^h y^i z$ , and we have

$$ \begin{align*} &F = z^2 u_1 + x^3 + y^3 + x^4 y u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ S_1 &= \{(6,0,0), (3,3,0), (0,6,0)\}, \qquad S_2 = \{(9,0)\}, \qquad S_3 = \{(0,9), (1,7), (2,5), (3,6)\}, \end{align*} $$

with $a_{hij}$ , $b_{hi}$ , $c_{hi}$ , and $u_1,u_2$ as in the previous case. By replacing y with $y + t x^2$ , we eliminate $a_{600}$ . By replacing x with $x + t' y^2$ and F with $(1 + t" y^3) F$ , we eliminate $a_{330}$ and $a_{060}$ . By replacing F with $(1 + x^2 y^2 c_{25} + x y^4 c_{17} + y^6 c_{09}) F$ , then with $(1 + x^6 b_{90} + x^3 y^3 c_{36}) F$ , we eliminate all $b_{hi}$ and $c_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ .

Hereafter, assume $p = 2$ and that $F_2$ is a square. If $\sqrt {F_2}$ is homogeneous, then we may assume $F_2 = x^2$ , we may assume $F_3$ has $y^2 z$ or $z^3$ , and then we have $(a,b,c) = (1,1,0)$ , and hence $F \in k[[x^2,xy,y^2,z]]$ . If $\sqrt {F_2}$ is not homogeneous, then we may assume $F_2 = x^2 + z^2$ and $a = 1$ and $c = 0$ , and then again we have $(a,b,c) = (1,1,0)$ , and hence $F \in k[[x^2,xy,y^2,z]]$ , and $F_3$ has $x y z$ , $y^2 z$ , $z^3$ , or $x^2 z$ .

Assume ( $F_2$ is $x^2$ or $x^2 + z^2$ and) $F_3$ contains $y^2 z$ . Furthermore, since $F \notin (x,y)^2$ , we see that F has $z^l$ ( $l \geq 2$ ). We have $F \equiv x^2 + y^2 z + z^l \pmod {(x^4, x^3 y, x^2 y^2, x y^3, y^4, x^2 z, x y z, y^2 z^2, z^{l+1})}$ . Write $F = F_0(x^2, y^2, z) + xy F_1(x^2, y^2, z)$ . Then there exist unique $f, g \in k[[z]]$ such that $F_0 \in (x^2 - f(z)^2, y^2 - g(z)^2)$ , and they satisfy $l = \min \{2 \operatorname {\mathrm {ord}}_z(f), 2 \operatorname {\mathrm {ord}}_z(g) + 1\}$ . If l is even, then, by replacing y with $y - x g/f$ , we may assume $g = 0$ . If l is odd, then, by replacing x with $x - y f/g$ , we may assume $f = 0$ . We eliminate $a_{hij}$ with $h \geq 2$ , $(h,i,j) \neq (2,0,0)$ , by replacing F with $(1 + a_{hij} x^{h-2} y^i z^j) F$ , and $a_{hij}$ with $i \geq 2$ , $(h,i,j) \neq (0,2,0), (0,2,1)$ , by replacing z with $z + a_{hij} x^h y^{i-2} z^j$ . We obtain $F = x^2 + y^2 z + z^l u(z) + x y e(z)$ , where $e(z) \in k[[z]]$ and $u(z) \in k[[z]]^*$ . We have $e(z) \neq 0$ , since if $e(z) = 0$ , then $F = F_0 \in ((x - f(z))^2, (y - g(z))^2)$ , which is absurd. Write $e(z) = z^k v(z)$ , $k \geq 1$ , and $v(z) \in k[[z]]^*$ . It is $D_{2k+l}^{\lfloor l/2 \rfloor }$ . If l is even, then, since $F_0 \in (x^2 - f(z)^2, y^2 - g(z)^2)$ and $g = 0$ , we have $z^l u(z) = f(z)^2$ and hence $u(z)$ is a square, and then by replacing x with $u(z)^{1/2} x$ and by replacing F with a unit multiple, we obtain $F = x^2 + y^2 z u'(z) + z^l + x y e(z)$ for some $u'(z) \in k[[z]]^*$ . Similarly, if l is odd, then, since $f = 0$ , we have $z^l u(z) = z g(z)^2$ and hence $u(z)$ is a square, and then (by replacing y) we obtain $F = x^2 u'(z) + y^2 z + z^l + x y e(z)$ . By replacing $x,y,z,F$ with unit multiples, we can achieve $u' = v = 1$ .

Assume $F_2 = x^2$ and $F_3$ has $z^3$ but no $y^2z$ . To define an RDP, F must have $y^4$ and must have $xyz$ or $xy^3$ . If F has $xyz$ , then it is $E_7^3$ . We have

$$ \begin{align*} &\qquad \quad F = x^2 + y^4 + z^3 u_1+ x y z u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j \\ &\qquad \quad + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i + \sum_{(h,i) \in S_4} d_{hi} x^h y^i, \\ S_1 &= \{(3,1,0), (1,3,0), (1,5,0), (0,2,2), (2,0,1), (2,0,2), (0,4,1), (0,4,2)\}, \\ &\quad S_2 = \{(4,0), (5,1)\}, \qquad S_3 = \{(0,6), (1,7)\}, \qquad S_4 = \{(2,2)\}, \end{align*} $$

where $a_{hij} \in k$ , $b_{hi} = \sum _{j = 0}^{2} b_{hij} z^j$ with $b_{hij} \in k[[x^2]]$ , $c_{hi} = \sum _{j = 0}^{2} c_{hij} z^j$ with $c_{hij} \in k[[y^2]]$ , $d_{hi} \in k[[x^2, x y, y^2]]$ , and $u_1,u_2 \in k[[x^2, x y, y^2, z]]^*$ . We may assume $u_1, u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We replace z with $z + a_{310} x^2 + a_{130} y^2 + a_{150} y^4$ , x with $x + t y z$ (which eliminates $a_{022}$ ), y with $y + a_{201} x + a_{202} x z$ , and x with $x + a_{041} y^3 + a_{042} y^3 z$ , and thus eliminate all $a_{hij}$ ( $(h,i,j) \in S_1$ ). We replace F with $(1 + x^2 b_{40} + y^2 c_{06}) F$ , F with $(1 + x^3 y b_{51} + x y^3 c_{17}) F$ , and z with $z + d_{22} x y$ , and thus eliminate all $b_{hi}$ , $c_{hi}$ , and $d_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1= u_2 = 1$ .

Next, if F does not have $xyz$ but has $x y^3$ , then it is $E_8^3$ . We have

$$ \begin{align*} &F = x^2 + y^4 + z^3 u_1 + x y^3 u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ &\quad S_1 = \{(2,0,1), (2,0,2), (0,4,1), (0,2,2), (1,1,2), (0,6,0), (0,4,2), (0,6,1), (0,6,2), \\ &\qquad (2,2,0), (2,2,1), (2,2,2)\}, \qquad S_2 = \{(4,0), (3,1), (4,2))\}, \qquad S_3 = \{(0,8)\}, \end{align*} $$

with $a_{hij}$ , $b_{hi}$ , $c_{hi}$ , and $u_1,u_2$ as in the previous case. We may assume $u_1, u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We replace F with $(1 + a_{201} z + a_{202} z^2) F$ , x with $x + t y z$ and z with $z + t' y^2$ (which eliminates $a_{041}$ and $a_{022}$ ), z with $z + a_{112} x y$ , x with $x + t" y^3$ (which eliminates $a_{060}$ ), x with $x + a_{042} y z^2 + a_{061} y^3 z$ , x with $x + a_{062} y^3 z^2$ , and y with $y + (a_{220} + a_{221} z + a_{222} z^2) x$ , and thus eliminate all $a_{hij}$ ( $(h,i,j) \in S_1$ ). We replace F with $(1 + x^2 b_{40} + x y b_{31} + x^2 y^2 b_{42} + y^4 c_{08}) F$ , and thus eliminate all $b_{hi}$ and $c_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ .

Assume $F_2 = x^2 + z^2$ and $F_3$ does not have $y^2 z$ and has $xyz$ . F moreover needs $x y^i$ , $y^i z$ , or $y^l$ . Replacing z with $z + xy^{i-1}$ (resp. x with $x + y^{i-1}z$ ), we may assume that there are no $x y^i$ (resp. $y^i z$ ) of low degree. Thus, we have

$$ \begin{align*} &\qquad\qquad\qquad\qquad\quad F \equiv x^2 + z^2 + xyz + y^{2n-2} \\ &\pmod{(x^4, x^3 y, x^2 y^2, x y^{m} (m> n), y^{2n}, x^2 z, x y z^2, y^2 z^2, y^{m} z (m > n), z^3)}, \end{align*} $$

$n \geq 3$ , and this is $D_{2n}^{n-1}[*]$ . We eliminate $x^h y^i$ ( $h ,i \geq 1$ , $(h, i) \neq (1, 1)$ ) by replacing z with $z + a_{hi0} x^{h-1} y^{i-1}$ , and $x^h z^j$ and $y^i z^j$ similarly. We eliminate $x^h$ , $y^i$ , $z^j$ ( $h \geq 3$ , $i \geq 2n-1$ , $j \geq 3$ ) by replacing F with a unit multiple. We obtain $F = x^2 + z^2 + xyz u + y^{2n-2}$ for some $u \in k[[x^2, x y, y^2, z]]^*$ , and we can achieve $u = 1$ .

Assume $F_2 = x^2 + z^2$ and $F_3$ does not have $y^2 z$ nor $xyz$ . By replacing F with $(1 + a_{201} x^2 z)^{-1} F$ , we may assume that F does not have $x^2 z$ . Then F has $z^3$ and F moreover needs $xy^3$ , and then it is $E_7^2$ . We have

$$ \begin{align*} &F = x^2 + z^2 + z^3 u_1 + x y^3 u_2 + \sum_{(h,i,j) \in S_1} a_{hij} x^h y^i z^j + \sum_{(h,i) \in S_2} b_{hi} x^h y^i + \sum_{(h,i) \in S_3} c_{hi} x^h y^i, \\ & \ \ S_1 = \{(0,4,0), (0,4,1), (0,6,0), (0,2,2), (3,1,0), (2,2,0), (1,1,2), (4,0,0), (2,0,2) \}, \\ &\qquad\qquad\qquad\qquad S_2 = \{(4,0), (3,1), (2,2)\}, \qquad S_3 = \{(0,4)\}, \end{align*} $$

with $a_{hij}$ , $b_{hi}$ , $c_{hi}$ , and $u_1,u_2$ as in the case of $E_7^3$ , and moreover $b_{hi} \in (x^2, z)$ and $c_{hi} \in (y^4, y^2 z, z^2)$ . We may assume $u_1, u_2 \equiv 1 \pmod {\mathfrak {m}}$ . We replace z with $z + a_{040}^{1/2} y^2$ , x with $x + a_{041} y z$ , x with $x + t y^3$ (which eliminates $a_{060}$ ), F with $(1 + a_{022} y^2) F$ , y with $y + t' x$ and z with $z + t" x y$ and F with $(1 + t"' x y) F$ (which eliminates $a_{310}$ , $a_{220}$ , and $a_{112}$ ), z with $z + t"" x^2$ and F with $(1 + t""' x^2) F$ (which eliminates $a_{400}$ and $a_{202}$ ), and thus eliminate all $a_{hij}$ ( $(h,i,j) \in S_1$ ). We replace x with $x + y c_{04}$ and F with $(1 + b_{40} x^2 + b_{31} x y + b_{22} y^2) F$ to eliminate all $b_{hi}$ and $c_{hi}$ . We replace $x,y,z,F$ with suitable multiples and achieve $u_1 = u_2 = 1$ .

4.3 $\mu _n$ -actions on RDPs

In this section, we classify $\mu _n$ -actions on RDPs under each of the following assumptions.

  • w is not fixed by $\mu _n$ (Proposition 4.12).

  • $n = p^e$ , and the subgroup scheme $\mu _p$ fixes w and is symplectic (Proposition 4.13).

  • w is fixed by $\mu _n$ and the action is symplectic (Proposition 4.14).

In Propositions 4.13 and 4.14, we use the convention that a smooth point is of type $A_0$ .

Lemma 4.11. Let X be a k-scheme equipped with a $\mu _{p^2}$ -action. Let $\pi _1 \colon X \to X_1 = X / \mu _p$ be the quotient morphism by the action of the subgroup scheme $\mu _p \subset \mu _{p^2}$ . If $w \in X$ is non-fixed by the action of $\mu _p$ , then $\pi _1(w) \in X_1$ is non-fixed by the action of $\mu _{p^2}/\mu _p$ .

Proof. Let $\mathcal {O}_{X,w} = B = \bigoplus _{i \in \mathbb Z/p^2\mathbb Z} B_i$ be the corresponding decomposition. Since w is non-fixed by $\mu _p$ , there exists $y \in \mathfrak {m}_{w} \subset B$ with $1 + y \in \bigoplus _{i \equiv 1 \pmod {p}} B_i$ (Lemma 2.9). Then, since $y^p \in \mathfrak {m}_{\pi _1(w)} \subset \mathcal {O}_{X_1, \pi _1(w)}$ satisfies $1 + y^p = (1 + y)^p \in B_p$ , we conclude by Proposition 2.8 that $\pi _1(w)$ is non-fixed by $\mu _{p^2}/\mu _p$ .

Suppose X is a scheme equipped with a $\mu _n$ -action, $n = p^e r$ with $p \nmid r$ , and $w \in X$ is a closed point fixed by $\mu _r \subset \mu _n$ . Let f be the maximal integer with $0 \leq f \leq e$ such that the subgroup scheme $\mu _{p^f} \subset \mu _{p^e} \subset \mu _n$ fixes w. We say that $\mu _{p^f r}$ is the stabilizer of w and denote it by $\operatorname {\mathrm {Stab}}(w)$ .

Proposition 4.12. Let